Evaluating multiphase fluid flow in a wellbore using temperature and pressure measurements

ABSTRACT

A system, method and program product for analyzing multiphase flow in a wellbore. A system is provided that includes: an input system for receiving pressure and temperature readings from a pair of sensors located in the wellbore; a computation system that utilizes a flow analysis model to generate a set of wellbore fluid properties, wherein the set of wellbore fluid properties includes at least one of: a fluid mixture value, a phase velocity value, a flow rate, a mixture density, a mixture viscosity, a fluid holdup, and a slip velocity; and a system for outputting the wellbore fluid properties.

FIELD OF THE INVENTION

This disclosure relates to evaluating fluid flow in an oil or gas well,and more particularly relates to a system and method of evaluatingmultiphase fluid flow in a wellbore using temperature and pressuremeasurements.

BACKGROUND OF THE INVENTION

Reliable and accurate downhole temperature and pressure measurementshave been available in the petroleum industry for the past severalyears. Permanent downhole pressure monitoring equipment has now beeninstalled in a number of producing basins around the world, withsuccessful measurement operations exceeding five or more years at thistime. Downhole permanent temperature measurements have also become morecommon, with both conventional or fiber optic thermal measurementscurrently available for most reservoir conditions. While continuouspressure and temperature readings provide an important part ofunderstanding oil and gas production, quantitative information musttypically be obtained using other types of data.

For example, the quantitative evaluation of the production or injectionprofile in an oil and/or gas well has traditionally involved the use ofproduction log measurements of flow rate, pressure, density, and fluidholdup to derive estimates of the wellbore fluid mixture phasevelocities, densities, pressure distributions, and completed intervalinflow or outflow contributions. Modern production logs can be used inmany situations to obtain the necessary measurements that are requiredto perform these quantitative computations. The measurements made inthese cases however are periodic and reflect the wellbore fluidinflows/outflows at the time that the production log was run.Unfortunately, the known art does not provide a solution to obtaincontinuous or real-time quantitative measurements and evaluations usingdownhole pressure and temperature readings obtained from a plurality ofsensors in the wellbore.

SUMMARY OF THE INVENTION

The present invention relates to a system, method and program productthat provides a computational model and evaluation technique for usingarray pressure and temperature measurements obtained in a flow conduitto evaluate the phase flow rates and velocities, fluid phase holdup,slip velocities between fluid phases, and mixture density and viscosity.These values can then be used, for instance, to quantify the inflow andoutflow contributions of completed zones in a wellbore.

In one embodiment, there is a system for analyzing multiphase flow in awellbore, comprising: an input system for receiving pressure andtemperature readings from a pair of sensors located in the wellbore; acomputation system that utilizes a flow analysis model to generate a setof wellbore fluid properties from the pressure and temperature readings,wherein the set of wellbore fluid properties includes at least one of: afluid mixture value, a phase velocity value, a flow rate, a mixturedensity, a mixture viscosity, a fluid holdup, and a slip velocity; and asystem for outputting the wellbore fluid properties.

In a second embodiment, there is a method for analyzing multiphase flowin a wellbore, comprising: obtaining pressure and temperature readingsfrom a pair of sensors located in the wellbore; utilizing a flowanalysis model to generate a set of wellbore fluid properties from thepressure and temperature readings, wherein the set of wellbore fluidproperties includes at least one of: a fluid mixture value, a phasevelocity value, a flow rate, a mixture density, a mixture viscosity, afluid holdup, and a slip velocity; and outputting the wellbore fluidproperties.

In a third embodiment, there is a computer readable medium for storing acomputer program product, which when executed by a computer systemanalyzes multiphase flow in a wellbore, comprising: program code forinputting pressure and temperature readings from a pair of sensorslocated in the wellbore; program code for implementing a flow analysismodel to generate a set of wellbore fluid properties from the pressureand temperature readings, wherein the set of wellbore fluid propertiesincludes at least one of: a fluid mixture value, a phase velocity value,a flow rate, a mixture density, a mixture viscosity, a fluid holdup, anda slip velocity; and program code for outputting the wellbore fluidproperties.

An advantage of this invention is the implementation of a quantitativeevaluation methodology for characterizing the temperature, pressure,wellbore fluid mixture density and viscosity, and fluid holdupdistributions in a wellbore using the temperature and pressuredistributions in the well. This is achieved by the development and useof a comprehensive multiphase capillary flow analysis model. The resultsprovide a reliable, accurate, and continuous characterization of thewellbore fluid flow properties such as pressure, temperature, mixturedensity, mixture viscosity, fluid phase holdup distributions, andcompleted zone inflow/outflow contributions.

This invention is directly applicable in wellbore environments andconditions in which modern production logging techniques may not bereadily accessible or may not be deployable as a result of the wellboregeometry, well depth, water depth, or other operational and economicalconsiderations.

The illustrative aspects of the present invention are designed to solvethe problems herein described and other problems not discussed.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features of this invention will be more readilyunderstood from the following detailed description of the variousaspects of the invention taken in conjunction with the accompanyingdrawings.

FIG. 1 depicts a computer system having a multiphase flow analysissystem in accordance with an embodiment of the present invention.

FIG. 2 depicts an embodiment of a multiphase flow analysis system thatprovides a contribution analysis in accordance with an embodiment of thepresent invention.

FIG. 3 depicts a graph showing a Fanning friction factor correlated withthe Reynolds number and relative roughness for single-phase flowsystems.

FIG. 4 depicts a graph showing a friction factor showing the transitionbetween the laminar and turbulent flow regimes in multiphase flowsystems.

The drawings are merely schematic representations, not intended toportray specific parameters of the invention. The drawings are intendedto depict only typical embodiments of the invention, and thereforeshould not be considered as limiting the scope of the invention. In thedrawings, like numbering represents like elements.

DETAILED DESCRIPTION OF THE INVENTION

Referring to the drawings, FIG. 1 depicts an overview of an illustrativesystem 11 for implementing aspects of the present invention. As shown, acomputer system 10 is provided that includes a multiphase flow analysissystem 18 for analyzing fluid characteristics flowing through a wellbore34. Also provided are at least two sensors 30, 32 placed within thewellbore to provide multipoint pressure and temperature readings.

Multiphase flow analysis system 18 includes a pressure and temperatureinput system 20 for obtaining pressure and temperature readings fromeach sensor 30, 32 in a continuous, as needed, or periodic manner. Alsoincluded is a computation system 22 that utilizes a flow analysis model24 for computing wellbore fluid properties, including one or more of:(1) the fluid mixture; (2) phase velocities; (3) flow rates; (4) mixturedensity; (5) mixture viscosity; (6) fluid holdups; and (7) estimates ofthe slip velocities between the wellbore liquid and gas phases andbetween the oil and water phases, if those phases are present in thesystem. Wellbore fluid properties 28 may be computed and outputted byoutput system 29 in an “on-demand” manner, i.e., continuously, asneeded, periodically, in real-time, etc. It is also possible to outputthe wellbore fluid properties when pre-selected system conditions arereached, such as anomalous incidents or trends, conditions exceedingthresholds, etc. A description of the flow analysis model 24 and how thecomputations may be implemented is provided below.

Also included in multiphase flow analysis system 18 is a contributionanalysis system 26 to quantitatively evaluate an oil or gas well withmultiple production or injection zones. For example, FIG. 2 depicts awell 50 having multiple production zones that include a main branch 42,a first contribution branch 44, and a second contribution branch 46. Inthis case, multipoint measurements are obtained with sets of sensorsconfigured into a multipoint measurements array. In particular, thecomplex multi-branched well 50 is fitted with three sets of sensors (A,B, and C). Each set is strategically located to obtain contributionreadings from each different zone in the well. Namely, sensor set C islocated to obtain readings for main branch 42; sensor set B is locatedto obtain contribution readings from main branch 42 and firstcontribution branch 44; and sensor set A is located to obtaincontribution readings from main branch 42, first contribution branch 44,and second contribution branch 46.

A contribution analysis 40 may be obtained for each contribution branch44, 46 by subtracting all the downstream fluid property computations.For instance, by subtracting computation values obtained from sensor setC from computation values obtained from sensor set B, a contributionanalysis 40 for the first contribution branch 44 can be obtained.Similarly, by subtracting computation values obtained from sensor sets Band C from computation values obtained from sensor set A, a contributionanalysis 40 for the second contribution branch 46 can be obtained.Contribution analysis 40 for main branch 42 is simply obtained fromsensor set C, which has no additional downstream contributions.

Note that each sensor set A, B, C may include more than two sensors inorder to provide redundancy. In this example, each set is shownincluding four sensors, e.g., set A includes sensors A1, A2, A3, and A4.This thus allows six different sensor pairs (e.g., A1-A2, A1-A3, A1-A4,A2-A3, A2-A4, A3-A4) to be used as a basis calculating wellbore fluidproperties. Any one or more of the sensor pairs may be used forevaluation purposes. While FIG. 2 depicts a well that has a main branchand first and second contribution branches, embodiments of the inventionmay also be used with a wellbore having only a main branch withdifferent inflow or outflow zones, such as a cased well having separateperforated intervals.

As noted in FIG. 1, computation system 22 provides a flow analysis model24 for generating wellbore fluid properties 28 for a sensor pair 30, 32.An explanation for how such properties may be obtained from temperatureand pressure readings from sensor pair 30, 32 begins with a review ofthe fundamental governing relationships that pertain to multiphase fluidflow in a tubular conduit (e.g., a wellbore). The following notation isused throughout the discussion.

Variable Description

-   D Flow conduit inside diameter-   f Fanning friction factor-   f_(o) Fraction of oil in liquid component of the system-   f_(w) Fraction of water in liquid component of the system-   g Gravitational acceleration-   H_(L) Liquid holdup-   h_(L) Elevation at end of flow conduit segment-   h₀ Elevation at start of flow conduit segment-   L Measured length of the flow conduit segment-   N_(Re) Reynolds number-   P_(L) Pressure at outlet end of flow conduit segment-   P₀ Pressure at start end of flow conduit segment-   q_(g) Insitu gas volumetric flow rate-   q_(L) Insitu liquid volumetric flow rate-   V Fluid average velocity in circular conduit-   V_(m) Wellbore fluid mixture superficial velocity, ft/s-   V_(sg) Gas superficial velocity, ft/s-   V_(sgL) Gas-liquid slip velocity, ft/s-   V_(sL) Liquid superficial velocity, ft/s-   V_(so) Oil superficial velocity, ft/s-   V_(sow) Oil-water slip velocity, ft/s-   V_(sw) Water superficial velocity, ft/s-   Y_(g) Gas holdup-   Y_(L) Liquid holdup-   Y_(o) Oil holdup-   Y_(w) Water holdup

Greek Description

-   α Wellbore deviation angle from vertical, deg-   ε Pipe roughness-   λ_(L) No-slip liquid holdup-   μ_(g) Gas dynamic viscosity, cp-   μ_(L) Liquid dynamic viscosity, cp-   μ_(m) Fluid mixture dynamic viscosity, cp-   μ_(o) Oil dynamic viscosity, cp-   μ_(w) Water dynamic viscosity, cp-   υ_(m) Fluid mixture kinematic viscosity, cp-cu ft/lbs-   ρ_(g) Gas density, lbs/cu ft-   ρ_(L) Liquid density, lbs/cu ft-   ρ_(m) Fluid mixture density, lbs/cu ft-   ρ_(o) Oil density, lbs/cu ft-   ρ_(osg) Oil density, g/cc-   ρ_(w) Water density, lbs/cu ft-   ρ_(wsg) Water density, g/cc

One of the fundamental parameters that can be used to quantify andcorrelate the level of inertial to viscous forces in a fluid flowing ina circular conduit is the Reynolds number. This dimensionless parameteris defined in Eq. 1.

$\begin{matrix}{N_{Re} = \frac{{DV}_{m}}{\upsilon_{m}}} & (1)\end{matrix}$

The kinematic viscosity of a fluid mixture appearing in Eq. 1 is definedas the ratio of the dynamic fluid viscosity to its density. Thisrelationship is expressed mathematically in Eq. 2.

$\begin{matrix}{\upsilon_{m} = \frac{\mu_{m}}{\rho_{m}}} & (2)\end{matrix}$

The general relationship that describes the pressure loss exhibited dueto fluid flow in a circular tubular conduit is given by Fanning'sequation. Note that gravitational effects have been included in thisexpression.

$\begin{matrix}{\frac{P_{0} - P_{L} + {\rho_{m}{g\left( {h_{0} - h_{L}} \right)}}}{L} = \frac{2\rho_{m}{fV}_{m}^{2}}{D}} & (3)\end{matrix}$

Substitution of Eqs. 2 and 3 into Eq. 1 results in expression that canbe used to correlate the Reynolds number and friction factor to theconduit dimensions, the pressure loss over a given length of conduit,and the physical properties of the fluid flowing in the conduit. Notethat the relationship given in Eq. 4 is explicitly independent of thefluid velocity, except that the effect of this parameter is implicitlymanifested in the fluid flow problem in the form of the Reynolds number.

$\begin{matrix}\begin{matrix}{{N_{Re}\sqrt{f}} = \frac{D^{\frac{3}{2}}{\rho_{m}^{\frac{1}{2}}\left\lbrack {P_{0} - P_{L} + {\rho_{m}{g\left( {h_{0} - h_{L}} \right)}}} \right\rbrack}^{\frac{1}{2}}}{\sqrt{2L}\mu_{m}}} \\{= \frac{{D^{\frac{3}{2}}\left\lbrack {P_{0} - P_{L} + {\rho_{m}{g\left( {h_{0} - h_{L}} \right)}}} \right\rbrack}^{\frac{1}{2}}}{\sqrt{2L\; \rho_{m}}\upsilon_{m}}}\end{matrix} & (4)\end{matrix}$

The Fanning friction factor encountered in Eqs. 3 and 4 is a function ofthe Reynolds number and the relative roughness of the conduit in whichthe flow occurs. The Fanning friction factor is correlated with theReynolds number and relative roughness as presented in FIG. 3. Thisfriction factor correlation is generally considered to be applicable tosingle-phase fluid flow. Note that there is an unstable transitionregime for Reynolds numbers in the range of 2,000 to 3,000. For Reynoldsnumbers below about 2,000, laminar flow conditions exist. For Reynoldsnumbers greater than approximately 3,000, turbulent flow conditions aregenerally considered to prevail.

Based upon gas-liquid experimental data, the friction factor that isapplicable for multiphase flow generally tends to have a smooth,continuous transition between the laminar and turbulent flow regimes.This transition regime behavior is depicted in FIG. 4. This transitionregime behavior was found to be valid for the typical range of piperelative roughness values that are commonly found in commerciallyavailable oilfield tubular goods

$\left( {\frac{ɛ}{D} \leq 0.004} \right).$

Note that in this case, the transition regime is a smooth transitionthat deviates from that of laminar flow at a Reynolds number ofapproximately 1,000, characterized by the inertial-turbulent frictionfactor values at higher Reynolds numbers.

The Fanning friction factor that corresponds to the laminar flow regimein FIGS. 3 and 4 (N_(Re)<2000 and N_(Re)<1000, respectively) can bedescribed mathematically with the relationship given in Eq. 5.

$\begin{matrix}{f = \frac{16}{N_{Re}}} & (5)\end{matrix}$

The Fanning friction factor that corresponds to the turbulent flowregime (N_(Re)>3,000) can be accurately evaluated using the relationshipdescribed in Colebrook, C. F.: “Turbulent Flow in Pipes, with ParticularReference to the Transition Region Between the Smooth and Rough PipeLaws,” J. Inst. Civil Engs., London, (1938-1939).

The evaluation of this expression requires an iterative numericalsolution procedure and is presented in Eq. 6.

$\begin{matrix}{\frac{1}{\sqrt{f}} = {{- 4}{\log \left( {{0.269\frac{ɛ}{D}} + \frac{1.255}{N_{Re}\sqrt{f}}} \right)}}} & (6)\end{matrix}$

In addition to the governing fluid flow relationships presented thusfar, a conservation of mass relationship for the fluids present in thesystem can also be defined. The fluid mixture density is generallycomputed in multiphase flow analyses in the manner depicted in Eq. 7.However, an alternate form of this relationship for flow in horizontalcircular conduits is described in Dukler, A. E.: “Gas-Liquid Flow inPipelines,” AGA, API, Vol. I, Research Results, (May 1969). Thatexpression is given in Eq. 8.

$\begin{matrix}{\rho_{m} = {{\rho_{L}Y_{L}} + {\rho_{g}\left( {1 - Y_{L}} \right)}}} & (7) \\{\rho_{m} = {{\rho_{L}\left( \frac{\lambda_{L}^{2}}{Y_{L}} \right)} + {\rho_{g}\left( \frac{\left( {1 - \lambda_{L}} \right)^{2}}{1 - Y_{L}} \right)}}} & (8)\end{matrix}$

The no-slip liquid holdup is utilized in Dukler's alternate fluidmixture density relationship. The no-slip liquid holdup is defined inEq. 9.

$\begin{matrix}{\lambda_{L} = \frac{q_{L}}{q_{L} + q_{g}}} & (9)\end{matrix}$

In a similar manner, the fluid mixture dynamic viscosity can beevaluated by various means. Hagedorn, A. R. and Brown, K. E.: “TheEffect of Liquid Viscosity in Vertical Two-Phase Flow,” JPT, (Feb.1964), 203, suggested that the fluid mixture viscosity in a multiphaseflow system should be evaluated in the manner given by Eq. 10.

$\begin{matrix}{\mu_{m} = {\mu_{L}^{Y_{L}}\mu_{g}^{1 - Y_{L}}}} & (10)\end{matrix}$

The fluid mixture dynamic viscosity has been more commonly estimated inprevious investigations of multiphase fluid flow using a holdup-weightedcombination of the liquid and gas viscosities, given by Eq. 11.

μ_(m)=μ_(L) Y _(L)+μ_(g)(1−Y _(L))   (11)

A relationship for the fluid mixture dynamic viscosity that is identicalto that given in Eq. 11 has been proposed, except that the no-slipliquid holdup is the weighting parameter used in those analyses ratherthan the slip-adjusted liquid holdup.

μ_(m)=μ_(L)λ_(L)+μ_(g)(1−λ_(L))   (12)

Where required, the kinematic viscosity (Eq. 3) can be evaluated usingthe fluid mixture density obtained with Eqs. 7 or 8 and dynamic fluidmixture viscosity evaluated with Eqs. 10, 11, or 12. An alternativeapproach is to evaluate the kinematic viscosity of the fluid mixture ina manner analogous to that used for the holdup-weighted mixture densityand viscosity. This expression is given in Eq. 13.

$\begin{matrix}{\upsilon_{m} = {{\frac{\mu_{L}}{\rho_{L}}Y_{L}} + {\frac{\mu_{g}}{\rho_{g}}\left( {1 - Y_{L}} \right)}}} & (13)\end{matrix}$

Regardless of the particular fluid mixture density and viscosityrelationship used in the analysis, most of the previous investigationsof multiphase flow in pipe have evaluated the liquid density and dynamicviscosity of oil and water mixtures using the relationships given inEqs. 14 and 15. Note that other mixture viscosity models may be used inthe analysis such a medium emulsion model for oil-water vertical flowsystems.

ρ_(L)=ρ_(o)f_(o)+ρ_(w)f_(w)   (14)

μ_(L)=μ_(o)f_(o)+μ_(w)f_(w)   (15)

Typically when these relationships for computing the liquid density anddynamic viscosity of oil-water systems are used, the fraction of oil andwater are often evaluated assuming no-slip conditions. However, asimilar analysis could also be performed using an appropriate sliprelationship between the water and the less dense oil phase in thesystem without any loss in generality. In addition, the slip velocityrelationship between the oil and water phases in a two-phase liquid flowsystem can be reliably determined using Eq. 16. An illustrativeembodiment provided herein utilizes this relationship (Eq. 16) for theoil-water slip velocity for wellbore inclinations up to about 70degrees, but the invention may also use other applicable oil-water slipvelocity correlations. This disclosure includes but is not limited tothe use of only a single oil-water slip velocity relationship in theinvention.

$\begin{matrix}{V_{sow} = {0.6569\left( {\rho_{wsg} - \rho_{osg}} \right)^{0.25}{\exp \left\lbrack {0.788\mspace{11mu} {\ln \left( \frac{1.85}{\rho_{wsg} - \rho_{osg}} \right)}\left( {1 - \frac{Y_{w}}{Y_{L}}} \right)} \right\rbrack}\left( {1 + {0.04\alpha}} \right)}} & (16)\end{matrix}$

The fundamental definition of the slip velocity between the oil andwater phases in a two-phase oil-water system is given by Eq. 17. It isnoted that the slip velocity between the oil and water phases is simplythe difference between the average velocities of the oil and waterphases. Note that when the definition of the slip velocity between theoil and water phases (two-phase relationship) is applied to athree-phase analysis (as is considered in this invention), the holdupsof the oil and water phases must be normalized by the liquid holdup inthe three-phase system. This normalization of the oil and water phaseholdups to the liquid holdup (oil+water) in a three-phase system ispresented in Eq. 17.

$\begin{matrix}{V_{sow} = {\frac{V_{so}Y_{L}}{Y_{o}} - \frac{V_{sw}Y_{L}}{Y_{w}}}} & (17)\end{matrix}$

A similar slip velocity relationship exists between the gas and liquidphases in a multiphase system. An accurate and reliable correlation forestimating the slip velocity between the gas and liquid phases in amultiphase system is given in Eq. 18. Other gas-liquid slip velocityrelationships may also be used in the computational analysis describedin this invention disclosure. While the gas-liquid slip velocityrelationship given in Eq. 18 provides an illustrative embodiment, theuse of other applicable gas-liquid slip velocity correlations may alsobe utilized and fall within the scope of this invention.

V _(sgL)=[(0.95−Y _(g) ²)^(0.5)+0.025](1+0.04α)   (18)

The fundamental definition of the slip velocity relationship between thegas and liquid phases in a multiphase system is given by Eq. 19.

$\begin{matrix}{V_{sgL} = {\frac{V_{sg}}{Y_{g}} - \frac{V_{sL}}{Y_{L}}}} & (19)\end{matrix}$

The liquid mixture superficial velocity in a multiphase system is thesum of the oil and water superficial velocities.

V _(sL) =V _(so) +V _(sw)   (20)

The sum of the holdups of each of the fluid phases must total to 1, thesum of all of the fluids in the system.

1=Y _(o) +Y _(w) +Y _(g) =Y _(L) +Y _(g)   (21)

The wellbore mixture fluid superficial velocity is the sum of thesuperficial velocities of each of the fluid phases present in thesystem.

V _(m) =V _(so) +V _(sw) +V _(sg) =V _(sL) +V _(sg)   (22)

The wellbore fluid mixture kinematic viscosity can be evaluated as thesum of the kinematic viscosities of each of the fluid phases and theirassociated fluid holdups.

$\begin{matrix}{\upsilon_{m} = \frac{{\mu_{o}Y_{o}} + {\mu_{w}Y_{w}} + {\mu_{g}Y_{g}}}{{\rho_{o}Y_{o}} + {\rho_{w}Y_{w}} + {\rho_{g}Y_{g}}}} & (23)\end{matrix}$

A final governing relationship that may be utilized to resolve theunknowns in the fluid flow problem is an expression relating the insitumixture density directly to the measured pressure and temperature, andthe composition of the fluids in the system. This relationship can be anequation-of-state, such as the model proposed by Peng and Robinson.Other equations-of state can also be used to evaluate the mixturedensity and fluid mixture viscosity at the insitu conditions oftemperature and pressure, for a given composition of wellbore fluid.

Evaluation of Single-Phase Flow Metering Parameters

In single-phase flow metering cases, the evaluation of the fluid flowparameters involves the solution of three equations for three unknownparameter values in the problem. In single-phase flow, the unknownparameters that must be determined in the analysis are the fluidsuperficial velocity (V_(si)), the Reynolds number (N_(Re)), and theFanning friction factor (f). The i subscript appearing on the phasesuperficial velocity and fluid properties in Eq. A-1 represents theindividual fluid phase (oil, gas, or water: i.e. o, g, or w). Thedefinition of the single-phase Reynolds number in conventional oilfieldunits is given in Eq. A-1.

$\begin{matrix}{N_{Re} = {\frac{124\mspace{11mu} D{V_{si}}}{\gamma_{i}} = \frac{124\mspace{11mu} D\; \rho_{i}{{Vsi}}}{\mu_{i}}}} & \left( {A\text{-}1} \right)\end{matrix}$

The Fanning friction factor for single-phase flow conditions isdetermined from FIG. 3. The Fanning friction is a function of theReynolds number and the relative pipe roughness.

For laminar flow conditions (N_(Re)<2,000), the Fanning friction factorgiven in FIG. 3 is defined by the relationship given in Eq. A-2.

$\begin{matrix}{f = \frac{16}{N_{Re}}} & \left( {A\text{-}2} \right)\end{matrix}$

Under turbulent flow conditions (N_(R)>3,000), the Fanning frictionfactor can be determined using the non-linear Colebrook-Whiterelationship given in Eq. A-3.

$\begin{matrix}{\frac{1}{\sqrt{f}} = {{- 4}{\log \left( {{0.269\frac{ɛ}{D}} + \frac{1.255}{N_{Re}\sqrt{f}}} \right)}}} & \left( {A\text{-}3} \right)\end{matrix}$

The final relationship that is used to resolve the unknowns in thesingle-phase flow metering problem is the capillary flow relationshipthat relates the pressure loss due to frictional and gravitationaleffects of flow in the conduit to the Reynolds number, fluid properties,and relative pipe roughness is given in Eq. A-4 using conventionaloilfield units.

$\begin{matrix}{{N_{Re}f^{1/2}} = \frac{1722.9\mspace{11mu} D^{3/2}{\rho_{i}^{1/2}\left\lbrack {P_{0} - P_{L} + {0.006945\rho_{i}L\; \cos \; \theta}} \right\rbrack}^{1/2}}{L^{1/2}\mu_{i}}} & \left( {A\text{-}4} \right)\end{matrix}$

The solution of these relationships for the three unknowns in theproblem may for example be accomplished using a non-linear root-solvingprocedure such as Secant-Newton. The parameter of variation in theroot-solving procedure is the superficial velocity (V_(si)). With thesingle-phase fluid physical properties (μ_(i), ρ_(i)) known as afunction of the pressure and temperature, the superficial velocity isused to determine the Reynolds number as defined in Eq. A-1, the Fanningfriction factor from FIG. 3 (or Eqs. A-2 or A-3), and the basis functionconstructed by rearranging the capillary flow relationship given in Eq.A-4.

Note that for a single-phase system, the oil-water and gas-liquid slipvelocities are of course equal to zero. The same is true of thesuperficial velocities and holdups of the fluid phases not present inthe single-phase system.

Evaluation of Oil-Water Two-Phase Flow Metering Parameters

The solution of two-phase flow metering computations using temperatureand pressure measurements described in this invention involve theresolution of a non-linear system of 10 independent relationships forthe 10 unknown parameters in the problem. This is true in oil-water,oil-gas, and water-gas two-phase flow metering analyses usingdistributed temperature and pressure measurements.

For an oil-water system, the unknowns that must be resolved in theanalysis are the oil and water holdups, the oil, water, and mixturesuperficial velocities, mixture density and dynamic viscosity, the slipvelocity between the oil and water phases, the Reynolds number andFanning friction factor, and pressure loss that occurs over the meteringlength of the flow conduit. Note that the gas holdup and superficialvelocity are equal to zero for an oil-water system, as is the gas-liquidslip velocity.

The first relationship that is used to construct the multiphase flowmetering analysis in oil-water systems is the holdup relationship givenin Eq. B-1.

1=Y _(o) +Y _(w)   (B-1)

The mixture density in oil-water two-phase flow can be defined by theexpression given in Eq. B-2.

ρ_(m)=ρ_(o) Y _(o)+ρ_(w) Y _(w)   (B-2)

The simultaneous solution of Eqs. B-1 and B-2 results in expressions forthe oil and water holdups, expressed in terms of the unknown mixturedensity.

$\begin{matrix}{Y_{o} = \frac{\rho_{w} - \rho_{m}}{\rho_{w} - \rho_{o}}} & \left( {B\text{-}3} \right) \\{T_{w} = \frac{\rho_{m} - \rho_{o}}{\rho_{w} - \rho_{o}}} & \left( {B\text{-}4} \right)\end{matrix}$

The two-phase oil-water flow mixture dynamic viscosity for non-emulsionfluid systems may be expressed by the relationship given in Eq. B-3.

μ_(m)=μ_(o) Y _(o)+μ_(w) Y _(w)   (B-5)

In terms of the unknown mixture density, the mixture viscosity isdefined as given in Eq. B-6.

$\begin{matrix}{\mu_{m} = \frac{{\mu_{o}\left( {\rho_{w} - \rho_{m}} \right)} + {\mu_{w}\left( {\rho_{m} - \rho_{o}} \right)}}{\rho_{w} - \rho_{o}}} & \left( {B\text{-}6} \right)\end{matrix}$

The mixture superficial velocity of an oil-water two-phase system is thesum of the superficial velocities of the oil and water phases.

V _(m) =V _(so) +V _(sw)   (B-7)

The superficial mass velocity of a two-phase oil-water flow stream isbest characterized using Eqs. B-2, B-7, and an equation-of-state. Anexpression that relates Eqs. B-2 and B-7 to the mass velocity is givenby Eq. B-8.

ρ_(m) V _(m)=(ρ_(o) Y _(o)+ρ_(w) Y _(w))(V _(so) +V _(sw))   (B-8)

Expressions for estimating the oil and water superficial velocitiesexpressed in terms of the mixture superficial velocity and density maybe obtained using the definition of the mass velocity given in Eq. B-8,in combination with an independent equation-of-state for computing themixture density using the temperature, pressure, and fluid composition.With the two measurements (differential pressure and temperature), twoparameters may be resolved in the oil-water two-phase system analysis,the mixture density and the velocity.

A slip velocity relationship that is applicable for oil and watermultiphase systems is presented in Eq. B-9, expressed in terms ofconventional oilfield units. This relationship relates the slip betweenthe oil and water phases to the differences in densities of the twofluids and the conduit inclination angle.

$\begin{matrix}{V_{sow} = {0.6569\; \left( \frac{\rho_{w} - \rho_{o}}{62.428} \right)^{0.25}{\exp \left\lbrack {{- 0.788}{\ln \left( \frac{115.5}{\rho_{w} - \rho_{o}} \right)}\left( \frac{\rho_{w} - \rho_{o}}{\rho_{w} - \rho_{o}} \right)} \right\rbrack}\left( {1 + {0.04\alpha}} \right)}} & \left( {B\text{-}9} \right)\end{matrix}$

The Reynolds number of oil-water two-phase flow in a circular conduit isgiven by Eq. B-10.

$\begin{matrix}{N_{Re} = \frac{124\rho_{m}D{V_{m}}}{\mu_{m}}} & \left( {B\text{-}10} \right)\end{matrix}$

Substitution of the oil and water superficial velocities, holdups (Eqs.B-3 and B-4), and the oil-water slip relationship into the definition ofthe Reynolds number (Eq. B-9), results in an expression for Reynoldsnumber that represents one component of the root-solving procedure basisfunction.The resulting expression can be used in conjunction with the capillaryflow relationship for a two-phase oil-water system, given in Eq. B-11,to construct a basis function for a non-linear root-solving procedurewith the mixture density as the variable parameter.

$\begin{matrix}{N_{Re} = \frac{1722.9\mspace{11mu} D^{3/2}{{\rho_{m}^{1/2}\left( {\rho_{w} - \rho_{o}} \right)}\begin{bmatrix}{P_{0} - P_{L} +} \\{0.006945\; \rho_{m}L\; \cos \; \theta}\end{bmatrix}}^{1/2}}{\sqrt{Lf}\left\lbrack {{\mu_{o}\left( {\rho_{w} - \rho_{m}} \right)} + {\mu_{w}\left( {\rho_{m} - \rho_{o}} \right)}} \right\rbrack}} & \left( {B\text{-}11} \right)\end{matrix}$

Once the system mixture density has been determined with theroot-solving procedure, the oil-water system slip velocity is evaluatedwith Eq. B-19, and the mixture velocity is evaluated with Eq. B-12.

$\begin{matrix}{V_{m} = \frac{\left\lbrack {{\rho_{w}\left( {\rho_{o} - \rho_{m}} \right)} + {\rho_{o}\left( {\rho_{w} - \rho_{m}} \right)}} \right\rbrack V_{sow}}{\rho_{w}^{2} - \rho_{o}^{2}}} & \left( {B\text{-}12} \right)\end{matrix}$

The Reynolds number can then be determined with Eq. B-10 and the Fanningfriction factor (f) is obtained with FIG. 4, which shows the Fanningfriction factor for multiphase flow systems (or with the laminar orColebrook-White turbulent flow relationships).

The oil and water phase superficial velocities are subsequentlyevaluated using expressions derived from the mixture and mass velocityrelationships (Eqs. B-7 and B-8), and the oil and water holdups areevaluated with Eqs. B-3 and B-4. The mixture dynamic viscosity can thenbe readily evaluated using Eq. B-5 or B-6.

Evaluation of Oil-Gas Two-Phase Flow Metering Parameters

Metering analyses using distributed temperature and pressuremeasurements in a two-phase oil-gas system involves the determination of10 unknown parameter values using 10 independent relationships, some ofwhich are non-linear and/or piece-wise continuous. The unknownparameters that must be resolved in an oil-gas two-phase system analysisare the following: oil and gas holdups, superficial velocities, themixture superficial velocity, density and viscosity, and the gas-liquidslip velocity, Reynolds number and Fanning friction factor. The waterholdup and superficial velocity in this case are equal to zero, as isthe oil-water slip velocity. Essentially with the two physicalmeasurements that are being made in this case, the differential pressureand the temperature, the mixture density and superficial velocity can beresolved.

The holdup relationship for a two-phase oil-gas system is given in Eq.C-1.

1=Y _(o) +Y _(g)   (C-1)

The mixture density is defined as in Eq. C-2.

ρ_(m)=ρ_(o) Y _(o)+ρ_(g) Y _(g)   (C-2)

The solution of these two relationships results in expressions for theoil and gas holdups in terms of the mixture density.

$\begin{matrix}{Y_{o} = \frac{\rho_{m} - \rho_{g}}{\rho_{o} - \rho_{g}}} & \left( {C\text{-}3} \right) \\{Y_{g} = \frac{\rho_{o} - \rho_{m}}{\rho_{o} - \rho_{g}}} & \left( {C\text{-}4} \right)\end{matrix}$

The mixture viscosity in a two-phase oil-gas system is generally definedin one of two ways, with the more common relationship given in Eq. C-5or with the Hagedorn-Brown model given in Eq. C-6.

μ_(m)=μ_(o) Y _(o)+μ_(g) Y _(g)   (C-5)

μ_(m)=μ_(o) ^(Y) ^(o) μ_(g) ^(Y) ^(g)   (C-6)

These expressions can be readily rewritten in terms of the oil and gasholdups given in Eqs. C-3 and C-4 as functions of the mixture density.

$\begin{matrix}{\mu_{m} = \frac{{\mu_{o}\left( {\rho_{m} - \rho_{g}} \right)} + {\mu_{g}\left( {\rho_{o} - \rho_{m}} \right)}}{\rho_{o} - \rho_{g}}} & \left( {C\text{-}7} \right) \\{\mu_{m} = {\mu_{o}^{(\frac{\rho_{m} - \rho_{g}}{\rho_{o} - \rho_{g}})}\mu_{g}^{(\frac{\rho_{o} - \rho_{m}}{\rho_{o} - \rho_{g}})}}} & \left( {C\text{-}8} \right)\end{matrix}$

The mixture superficial velocity of the two-phase system is defined inEq. C-9, with the superficial mass velocity being evaluated with Eq.C-10.

V _(m) =V _(so) +V _(sg)   (C-9)

ρ_(m) V _(m)=(ρ_(o) Y _(o)+ρ_(g) Y _(g))(V _(so) +V _(sg))   (C-10)

Solution of Eqs. C-9 and C-10, with substitution of the previouslydetermined relationships for the holdups (Eqs. C-3 and C-4), the oil andgas superficial velocities can be expressed in terms of the mixturedensity and superficial velocity. The mixture density in this case isbest characterized using an accurate equation-of-state to determine thedensities of the liquid and vapor phases in the system.The gas-liquid slip velocity relationship is defined for an oil-gastwo-phase system as shown in Eq. C-11.

$\begin{matrix}{{\frac{V_{sg}}{Y_{g}} - \frac{V_{so}}{Y_{o}}} = {\left\lbrack {\left( {0.95 - Y_{g}^{2}} \right)^{0.5} + 0.025} \right\rbrack \left( {1 + {0.04\alpha}} \right)}} & \left( {C\text{-}11} \right)\end{matrix}$

The Reynolds number of a two-phase oil-gas flow is determined using Eq.C-12.

$\begin{matrix}{N_{Re} = \frac{124\mspace{11mu} \rho_{m}D{V_{m}}}{\mu_{m}}} & \left( {C\text{-}12} \right)\end{matrix}$

Substitution of the mixture superficial velocity given by Eq. C-9 intothe Reynolds number relationship (Eq. C-12) yields an expression thatcan be used to construct a basis function for a root-solving procedureto evaluate the unknown parameter values in the oil-gas two-phase flowmetering problem.

Another expression for the Reynolds number can be obtained from thecapillary flow relationship that describes the pressure differential inthe flow conduit due to frictional and gravitational effects. Thisrelationship is given in Eq. C-13 and is used to complete theconstruction of the root-solving basis function used in the analysis.Note that the Fanning friction factor appearing in Eq. C-13 is obtainedfrom FIG. 4 as a function of the Reynolds number and the relative piperoughness, or by the solution of the laminar or turbulent flowrelationships that correspond to the graphical solution of the Fanningfriction factor.

$\begin{matrix}{N_{Re} = \frac{1722.9\mspace{11mu} D^{3/2}{{\rho_{m}^{1/2}\left( {\rho_{o} - \rho_{g}} \right)}\begin{bmatrix}{P_{0} - P_{L} +} \\{0.006945\rho_{m}L\; \cos \; \theta}\end{bmatrix}}^{1/2}}{\sqrt{Lf}\left\lbrack {{\mu_{o}\left( {\rho_{m} - \rho_{g}} \right)} + {\mu_{g}\left( {\rho_{o} - \rho_{m}} \right)}} \right\rbrack}} & \left( {C\text{-}13} \right)\end{matrix}$

The unknown parameter used as the variable of the root-solving procedurein this analysis is the mixture density. Once the mixture density hasbeen determined, the Reynolds number can be readily evaluated usingeither Eq. C-12 or C-13. The mixture superficial velocity is thenevaluated with Eq. C-9. The oil and gas phase holdups may be determinedwith Eqs. C-3 and C-4, followed by the mixture dynamic viscositycomputed with Eq. C-5.

A similar solution methodology can be developed for the alternatemixture viscosity relationship given in Eq. C-6, as that given when Eq.C-5 is used. The solution methodology developed in this invention isapplicable in general for all oil-gas two-phase flow cases. Substitutionfor an alternate dynamic viscosity or gas-liquid slip velocityrelationship is permitted in the analysis.

Evaluation of Water-Gas Two-Phase Flow Metering Parameters

In a water-gas two-phase flow metering system developed usingdistributed temperature and pressure measurements, the evaluation of the10 unknown parameters require the use of 10 independent functionalrelationships involving those parameters in order to resolve themultiphase flow metering problem. The unknown parameter values that mustbe determined from the metering analysis are the water and gas holdups,the water and gas superficial velocities, the mixture superficialvelocity, density and dynamic viscosity, the gas-liquid slip velocity,Reynolds number and Fanning friction factor. In a manner similar to thatdescribed previously for the other two-phase flow metering analyses, anon-linear root-solving procedure is required to resolve the unknowns ofthe problem. Note that in a two-phase water-gas flow metering analysis,the oil holdup and superficial velocity are equal to zero, as well as isthe oil-water slip velocity.

The holdup relationship that is applicable for a two-phase water-gasmetering analysis is given by Eq. D-1.

1=Y _(w) +Y _(g)   (D-1)

The mixture density of the water-gas two-phase flow is defined by Eq.D-2.

ρ_(m)=ρ_(w) Y _(w)+ρ_(g) Y _(g)   (D-2)

Simultaneous solution of Eqs. D-1 and D-2 results in expressions for thewater and gas holdups, expressed in terms of the fluid mixture densityof the water-gas system.

$\begin{matrix}{Y_{w} = \frac{\rho_{m} - \rho_{g}}{\rho_{w} - \rho_{g}}} & \left( {D\text{-}3} \right) \\{Y_{g} = \frac{\rho_{w} - \rho_{m}}{\rho_{w} - \rho_{g}}} & \left( {D\text{-}4} \right)\end{matrix}$

There are at least two fluid mixture viscosity relationships that can beused for characterizing the dynamic fluid viscosity in a water-gastwo-phase metering analysis. The more commonly used of these is therelationship given in Eq. D-5, with an alternate fluid mixture viscosityrelationship proposed by Hagedorn and Brown given in Eq. D-6.

μ_(m)=μ_(w) Y _(w)+μ_(g) Y _(g)   (D-5)

μ_(m)=μ_(w) ^(Y) ^(w) μ_(g) ^(Y) ^(g)   (D-6)

Application of the holdup relationships obtained in Eqs. D-3 and D-4 inthe mixture viscosity model given by Eq. D-5, results in a fluid mixtureviscosity relationship that is only a function of the unknown fluidmixture density.

$\begin{matrix}{\mu_{m} = \frac{{\mu_{w}\left( {\rho_{m} - \rho_{g}} \right)} + {\mu_{g}\left( {\rho_{w} - \rho_{m}} \right)}}{\rho_{w} - \rho_{g}}} & \left( {D\text{-}7} \right)\end{matrix}$

The mixture superficial velocity in a water-gas two-phase flow meteringanalysis is the sum of the water and gas phase superficial velocities.

V _(m) =V _(sw) +V _(sg)   (D-8)

The superficial mass velocity in the water-gas system can be evaluatedas defined in Eq. D-9.

ρ_(m) V _(m)=(ρ_(w) Y _(w)+ρ_(g) Y _(g))(V _(sw) +V _(sg))   (D-9)

Solution of Eqs. D-8 and D-9, with the definitions of the water and gasholdups previously obtained in Eqs. D-3 and D-4, the superficialvelocity of the water and gas phases can be expressed in terms of themixture density and superficial velocity.The gas-liquid slip velocity can be evaluated using the slip velocityrelationship presented in Eq. D-10.

$\begin{matrix}{{\frac{V_{sg}}{Y_{g}} - \frac{V_{sw}}{Y_{w}}} = {\left\lbrack {\left( {0.95 - Y_{g}^{2}} \right)^{0.5} + 0.025} \right\rbrack \left( {1 + {0.04\alpha}} \right)}} & \left( {D\text{-}10} \right)\end{matrix}$

The fluid mixture superficial velocity is given in Eq. D-8 and theReynolds number for a water-gas multiphase flow is defined by therelationship given in Eq. D-11.

$\begin{matrix}{N_{Re} = \frac{124\mspace{11mu} \rho_{m}D{V_{m}}}{\mu_{m}}} & \left( {D\text{-}11} \right)\end{matrix}$

Substitution of the results of mixture dynamic viscosity (Eq. D-7) andsuperficial velocity (Eq. D-8) in the Reynolds number relationshipresults in one component of the basis function for evaluating theunknowns in the multiphase metering problem.

The other component of the basis function (alternate Reynolds numberrelationship) is obtained from the capillary flow relationship thatrelates the pressure differential observed in flow in a conduit to thefrictional and gravitational effects.

$\begin{matrix}{N_{Re} = \frac{1722.9\mspace{11mu} D^{3/2}{{\rho_{m}^{1/2}\left( {\rho_{w} - \rho_{g}} \right)}\begin{bmatrix}{P_{0} - P_{L} +} \\{0.006945\mspace{11mu} \rho_{m}L\; \cos \; \theta}\end{bmatrix}}^{1/2}}{\sqrt{Lf}\left\lbrack {{\mu_{w}\left( {\rho_{m} - \rho_{g}} \right)} + {\mu_{g}\left( {\rho_{w} - \rho_{m}} \right)}} \right\rbrack}} & \left( {D\text{-}12} \right)\end{matrix}$

With the fluid mixture density obtained from the root-solving procedurejust described, the Reynolds number can then be determined using eitherEq. D-11 or D-12. With the two independent measurements available(differential pressure and temperature) two parameters of the problemcan be resolved. These are the mixture density and the superficialvelocity. A mixture density can be derived from the constituitiverelationships of the problem, including the capillary flow relationshipand the differential pressure measurements. The pressure and temperaturealso provides a means of computing the mixture density under theseconditions as a function of the fluid composition using an accurate androbust equation-of-state.

Evaluation of Three-Phase Flow Metering Parameters

The evaluation of the unknown metering parameters in a three-phasesystem (oil, gas, and water) using distributed temperatures andpressures is by far the most difficult to implement in a stablenumerical solution procedure due to the complex relationships betweenthe slip velocities, holdups, mixture viscosities, and superficialvelocities of the phases present in the system. The unknowns in thethree-phase metering analysis include the holdups of all three phases,their superficial velocities, as well as the mixture superficialvelocity, the mixture density, viscosity, Reynolds number and frictionfactor, in addition to the water-oil and gas-liquid slip velocities ofthe system. There are a total of 13 unknowns in the three-phase meteringanalysis problem. Therefore, a total of 13 independent relationships arerequired to properly resolve the unknowns in the three-phase flowmetering analysis using distributed temperature and pressuremeasurements.

As was demonstrated with the two-phase flow problems above, the holduprelationship is the first fundamental independent relationship that isused to construct the system of equations in the analysis. Thethree-phase holdup relationship is given by Eq. E-1.

1=Y _(o) +Y _(w) +Y _(g)   (E-1)

The fluid mixture density is defined in the three-phase analysis withEq. E-2.

ρ_(m)=ρ_(o) Y _(o)+ρ_(w) Y _(w)+ρ_(g) Y _(g)   (E-2)

The fluid mixture dynamic viscosity is commonly evaluated using themodel presented in Eq. E-3.

μ_(m)=μ_(o) Y _(o)+μ_(w) Y _(w)+μ_(g) Y _(g)   (E-3)

An alternate expression for estimating the fluid mixture dynamicviscosity has been proposed by Hagedorn and Brown. The Hagedorn andBrown fluid mixture viscosity model is given in Eq. E-4.

$\begin{matrix}{\mu_{m} = {\left( \frac{{\mu_{o}Y_{o}} + {\mu_{w}Y_{w}}}{Y_{o} + Y_{w}} \right)^{({Y_{o} + Y_{w}})}\mu_{g}^{Y_{g}}}} & \left( {E\text{-}4} \right)\end{matrix}$

One fluid mixture relationship that has been found to characterize thekinematic viscosity of the three-phase system reasonably well is givenby Eq. E-5. The kinematic viscosity is defined as the ratio of thedynamic viscosity to the fluid mixture density.

$\begin{matrix}{\gamma_{m} = {\frac{\mu_{m}}{\rho_{m}} = \frac{{\mu_{o}Y_{o}} + {\mu_{w}Y_{w}} + {\mu_{g}Y_{g}}}{{\rho_{o}Y_{o}} + {\rho_{w}Y_{w}} + {\rho_{g}Y_{g}}}}} & \left( {E\text{-}5} \right)\end{matrix}$

The simultaneous solution of Eqs. E-1 through E-5 can be used to developexpressions for the three fluid phase holdups and the mixture viscosity,expressed in terms of the unknown mixture density. The mixture kinematicviscosity is a sum of the kinematic viscosities of the individualphases, the gas holdup can then be evaluated as a function of themixture density and dynamic viscosity.

The water holdup may then be evaluated using Eq. E-6 as a function ofthe mixture density and viscosity, and the gas holdup. The oil phaseholdup can subsequently be computed from the fundamental holduprelationship given in Eq. E-1 using the results of the gas holdup andE-6.

$\begin{matrix}{Y_{w} = \frac{\rho_{m} - \rho_{o} - {Y_{g}\left( {\rho_{g} - \rho_{o}} \right)}}{\rho_{w} - \rho_{o}}} & \left( {E\text{-}6} \right)\end{matrix}$

The three-phase flow metering analysis solution procedure next addressesthe issue of the fluid phase and mixture superficial velocities and thetwo sets of two-phase slip velocity relationships that are required inthe analysis of a three-phase flow system. The slip velocityrelationships that are applicable for the oil and water phases in athree-phase analysis are given by Eqs. E-7 and E-8.

$\begin{matrix}{\mspace{79mu} {V_{sow} = {\frac{V_{so}\left( {Y_{o} + Y_{w}} \right)}{Y_{o}} - \frac{V_{sw}\left( {Y_{o} + Y_{w}} \right)}{Y_{w}}}}} & \left( {E\text{-}7} \right) \\{V_{sow} = {0.6569\left( \frac{\rho_{w} - \rho_{o}}{62.428} \right)^{0.25}{\exp\begin{bmatrix}{{- 0.788}\mspace{11mu} {\ln \left( \frac{115.5}{\rho_{w} - \rho_{o}} \right)}} \\\left( \frac{Y_{o}}{Y_{o} + Y_{w}} \right)\end{bmatrix}}\left( {1 + {0.04`\alpha}} \right)}} & \left( {E\text{-}8} \right)\end{matrix}$

The gas-liquid slip velocity relationships that are applicable inthree-phase flow analyses are presented in Eqs. E-9 and E-10.

$\begin{matrix}{V_{sgL} = {\frac{V_{sg}}{Y_{g}} - \frac{V_{so} + V_{sw}}{Y_{o} + Y_{w}}}} & \left( {E\text{-}9} \right) \\{V_{sgL} = {\left\lbrack {\left( {0.95 - Y_{g}^{2}} \right)^{0.5} + 0.025} \right\rbrack \left( {1 + {0.04\alpha}} \right)}} & \left( {E\text{-}10} \right)\end{matrix}$

The three-phase mixture superficial velocity is given by Eq. E-11.

V _(m) =V _(so) +V _(sw) +V _(sg)   (E-11)

The mass superficial velocity can best be characterized using arelationship such as the one given in Eq. E-12 and a value of themixture density derived from an accurate equation-of-state.

ρ_(m) V _(m)=(ρ_(o) Y _(o)+ρ_(w) Y _(w)+ρ_(g) Y _(g))(V _(so) +V _(sw)+V _(sg))   (E-12)

The solution of Eqs. E-7 through E-12 results in expressions for thephase and mixture superficial velocities and slip velocities that areonly functions of the previously determined fluid phase holdups anddynamic viscosity, all of which can be directly related to the fluidmixture density.One component of the root-solving procedure basis function is obtainedin the form of the Reynolds number, given by Eq. E-13.

$\begin{matrix}{N_{Re} = \frac{124\mspace{11mu} \rho_{m}D{V_{m}}}{\mu_{m}}} & \left( {E\text{-}13} \right)\end{matrix}$

Substitution into Eq. E-13 for the mixture superficial velocity (Eq.E-11) and dynamic viscosity (Eq. E-3) results in one component of thebasis function of the root-solving procedure used in the three-phaseflow metering analysis. The other component of the basis function usedin the root-solving procedure for evaluating the mixture density,satisfying all of the conditions and relationships in the three-phaseflow metering analysis, is obtained from the capillary flowrelationship. Rearranged in terms of the Reynolds number, thisrelationship is given in Eq. E-14. The Fanning friction factor in thisexpression is evaluated using FIG. 4 for the Reynolds number defined byEq. E-14.

$\begin{matrix}{N_{Re} = \frac{1722.9\mspace{11mu} D^{3/2}{{\rho_{m}^{1/2}\left( {\rho_{w} - \rho_{g}} \right)}\begin{bmatrix}{P_{0} - P_{L} +} \\{0.006945\mspace{11mu} \rho_{m}L\; \cos \; \theta}\end{bmatrix}}^{1/2}}{\sqrt{Lf}\left\lbrack {{\mu_{o}Y_{o}} + {\mu_{w}Y_{w}} + {\mu_{g}Y_{g}}} \right\rbrack}} & \left( {E\text{-}14} \right)\end{matrix}$

With the three-phase fluid mixture density resolved with the non-linearroot-solving mixture described herein, the unknown parameters in theproblem are recovered by back-substitution in the analysis procedure.For instance, the Reynolds number can be computed directly using themixture density and Eqs. E-13 or E-14. The mixture superficial velocityis determined with Eq. E-11 and the mixture dynamic viscosity isobtained with Eq. E-3. The oil-water system slip velocity can beevaluated using Eq. E-8 and the gas-liquid slip velocity can be computedwith Eq. E-10. The water phase superficial velocity can be evaluatedusing Eq. E-15 and the gas phase superficial velocity can be evaluatedwith Eq. E-16. The oil phase superficial velocity can then be determinedby rearranging Eq. E-11.

$\begin{matrix}{V_{sw} = {Y_{w}\left\lbrack {V_{m} - {V_{sgL}Y_{g}} - \frac{V_{sow}Y_{o}}{\left( {Y_{o} + Y_{w}} \right)^{2}}} \right\rbrack}} & \left( {E\text{-}15} \right) \\{V_{sg} = {Y_{g}\left\lbrack {V_{m} + {V_{sgL}\left( {Y_{o} + Y_{w}} \right)}} \right\rbrack}} & \left( {E\text{-}16} \right)\end{matrix}$

Example Computational Results

The results of an example computation of multiphase flow velocities,holdup, slip velocities, and mixture density and viscosity for apressure traverse in a vertical section of wellbore production tubingusing the computational methodology disclosed in this invention ispresented in the following discussion. The fluids considered in thistheoretical example include a 40° API hydrocarbon liquid (oil) with adensity of 45.923 lbs/cu ft and a dynamic viscosity of 0.487 cp,produced formation water with a salinity of 40,000 ppm that has adensity of 65.762 lbs/cu ft and a dynamic viscosity of 0.271 cp, and anatural gas mixture that has a density at downhole wellbore conditionsof 2.456 lbs/cu ft and a dynamic viscosity of 0.014 cp.

Simulated temperature and pressure measurements are modeled for twospatial positions in a vertical section of the wellbore for multiphaseflow metering purposes, at wellbore depths of 10,000 and 10,100 ft. Thetemperature in the wellbore at 10,000 ft of depth was assumed to be 240°F. and the flowing wellbore pressure at that depth was assumed to be1,000 psia. At 10,100 ft of depth, the corresponding temperature wasmodeled to be 241.8° F. and the wellbore flowing pressure was assumed tobe 1,025 psia. The production tubing (flow conduit) in this section ofthe wellbore in this example is 2 ⅜in OD tubing which has an internaldiameter of 1.995 inches and a relative roughness of 0.004.

An example of the output results obtained using a computer programconsisting of the computational methodology described in this inventiondisclosure is presented in the following summary table. Note that inthis synthetic example there is three-phase flow in the wellbore. Infact, there is upward flow of gas while there is fallback (downwardflow) of the hydrocarbon liquid (oil) and water phases. The Reynoldsnumber indicates that the flow conditions are in the transition flowregime range (not quite fully developed turbulent flow) and the pipefriction is relatively low due to the moderate Reynolds number and therelatively low relative roughness of the conduit.

The gas holdup obtained for these conditions indicates that gas occupies36% of the wellbore flow area, with water present in about 30%, and oiloccupying about 34% of the flow area or volume. The volumetric flowrates obtained in the analysis are presented in the summary tables aswell. Note that the gas volumetric flow rate includes the free gaspresent in the flowstream, as well as the solution gas dissolved in theoil and water phases at downhole conditions.

Wellbore Segment Computed Results: Oil holdup = 0.337 Water holdup =0.298 Gas holdup = 0.364 Oil superficial velocity = −0.009 ft/s Watersuperficial velocity = −0.123 ft/s Gas superficial velocity = 0.062 ft/sLiquid superficial velocity = −0.131 ft/s Mixture superficial velocity =−0.069 ft/s Oil-water slip velocity = 0.244 ft/s Gas-liquid slipvelocity = 0.378 ft/s Mixture density = 35.999 lbs/cu ft Mixture dynamicviscosity = 0.250 cp Mixture kinematic viscosity = 0.007 cp-cu ft/lbsReynolds number = 2456.4 Fanning friction factor = 0.01238 Oil flow rate= −2.54 STB/D Water flow rate = −38.84 STB/D Gas flow rate = 5.62 Mscf/D

Referring again to FIG. 1, it is understood that computer system 10 maybe implemented as any type of computing infrastructure. Computer system10 generally includes a processor 12, input/output (I/O) 14, memory 16,and bus 17. The processor 12 may comprise a single processing unit, orbe distributed across one or more processing units in one or morelocations, e.g., on a client and server. Memory 16 may comprise anyknown type of data storage, including magnetic media, optical media,random access memory (RAM), read-only memory (ROM), a data cache, a dataobject, etc. Moreover, memory 16 may reside at a single physicallocation, comprising one or more types of data storage, or bedistributed across a plurality of physical systems in various forms.

I/O 14 may comprise any system for exchanging information to/from anexternal resource. External devices/resources may comprise any knowntype of external device, including sensors 30, 32, a monitor/display,speakers, storage, another computer system, a hand-held device,keyboard, mouse, wireless system, voice recognition system, speechoutput system, printer, facsimile, pager, etc. Bus 17 provides acommunication link between each of the components in the computer system10 and likewise may comprise any known type of transmission link,including electrical, optical, wireless, etc. Although not shown,additional components, such as cache memory, communication systems,system software, etc., may be incorporated into computer system 10.

Access to computer system 10 may be provided over a network such as theInternet, a local area network (LAN), a wide area network (WAN), avirtual private network (VPN), etc. Communication could occur via adirect hardwired connection (e.g., serial port), or via an addressableconnection that may utilize any combination of wireline and/or wirelesstransmission methods. Moreover, conventional network connectivity, suchas Token Ring, Ethernet, WiFi or other conventional communicationsstandards could be used. Still yet, connectivity could be provided byconventional TCP/IP sockets-based protocol. In this instance, anInternet service provider could be used to establish interconnectivity.Further, communication could occur in a client-server or server-serverenvironment.

It should be appreciated that the teachings of the present inventioncould be offered as a business method on a subscription or fee basis.For example, a computer system 10 comprising a multiphase flow analysissystem 18 could be created, maintained and/or deployed by a serviceprovider that offers the functions described herein for customers. Thatis, a service provider could offer to provide wellbore fluid propertyinformation as described above.

It is understood that in addition to being implemented as a system andmethod, the features may be provided as a program product stored on acomputer-readable medium, which when executed, enables computer system10 to provide a multiphase flow analysis system 18. To this extent, thecomputer-readable medium may include program code, which implements theprocesses and systems described herein. It is understood that the term“computer-readable medium” comprises one or more of any type of physicalembodiment of the program code. In particular, the computer-readablemedium can comprise program code embodied on one or more portablestorage articles of manufacture (e.g., a compact disc, a magnetic disk,a tape, etc.), on one or more data storage portions of a computingdevice, such as memory 16 and/or a storage system, and/or as a datasignal traveling over a network (e.g., during a wired/wirelesselectronic distribution of the program product).

As used herein, it is understood that the terms “program code” and“computer program code” are synonymous and mean any expression, in anylanguage, code or notation, of a set of instructions that cause acomputing device having an information processing capability to performa particular function either directly or after any combination of thefollowing: (a) conversion to another language, code or notation; (b)reproduction in a different material form; and/or (c) decompression. Tothis extent, program code can be embodied as one or more types ofprogram products, such as an application/software program, componentsoftware/a library of functions, an operating system, a basic I/Osystem/driver for a particular computing and/or I/O device, and thelike. Further, it is understood that terms such as “component” and“system” are synonymous as used herein and represent any combination ofhardware and/or software capable of performing some function(s).

The block diagrams in the figures illustrate the architecture,functionality, and operation of possible implementations of systems,methods and computer program products according to various embodimentsof the present invention. In this regard, each block in the blockdiagrams may represent a module, segment, or portion of code, whichcomprises one or more executable instructions for implementing thespecified logical function(s). It should also be noted that thefunctions noted in the blocks may occur out of the order noted in thefigures. For example, two blocks shown in succession may, in fact, beexecuted substantially concurrently, or the blocks may sometimes beexecuted in the reverse order, depending upon the functionalityinvolved. It will also be noted that each block of the block diagramscan be implemented by special purpose hardware-based systems whichperform the specified functions or acts, or combinations of specialpurpose hardware and computer instructions.

Calibration Using Data From A Retrievable Production Logging Device

Embodiments of the inventive system, method, and program code can alsoutilize data obtained from a retrievable production logging device tocalibrate one or more of the generated wellbore fluid properties. Theseretrievable production logging devices are typically deployed in thewellbore on wireline, slickline, or coiled tubing. In the case of highlydeviated or horizontal wellbores, the production logging devices may bepushed into position using coiled tubing or stiff wireline cable or maybe pulled into position using a downhole tractor. Examples of the typesof retrievable production logging devices that may be used include theProduction Logging Tool, Memory PS Platform, Gas Holdup Optical SensorTool, and Flow Scanner Tool, all available from Schlumberger. Thecalibration process may involve the identification of or confirmationthat one or more sensors that are providing inaccurate downholemeasurements and elimination/rejection of the data provided by thesesensors. Alternatively, the wellbore fluid property generation processand/or results may be adjusted to either match or more closely reflectthe data obtained from the retrievable production logging device.

Although specific embodiments have been illustrated and describedherein, those of ordinary skill in the art appreciate that anyarrangement which is calculated to achieve the same purpose may besubstituted for the specific embodiments shown and that the inventionhas other applications in other environments. This application isintended to cover any adaptations or variations of the presentinvention. The following claims are in no way intended to limit thescope of the invention to the specific embodiments described herein.

1. A system for analyzing multiphase flow in a wellbore, comprising: aninput system for receiving pressure and temperature readings from a pairof sensors located in the wellbore; a computation system that utilizes aflow analysis model to generate a set of wellbore fluid properties fromthe pressure and temperature readings, wherein the set of wellbore fluidproperties includes at least one of: a fluid mixture value, a phasevelocity value, a flow rate, a mixture density, a mixture viscosity, afluid holdup, and a slip velocity; and a system for outputting thewellbore fluid properties.
 2. The system of claim 1, wherein the pair ofsensors are permanently mounted in the wellbore.
 3. The system of claim1, further comprising a contribution analysis system for analyzingpressure and temperature from a plurality of sensor pairs to providewellbore fluid properties for different inflow zones and/or branches ofa multiple production zone well.
 4. The system of claim 1, wherein theflow analysis model includes a single-phase flow model that utilizesthree equations to solve for a fluid superficial velocity (V_(si)), aReynolds number (N_(Re)), and a Fanning friction factor (f).
 5. Thesystem of claim 1, wherein the flow analysis model includes a two-phaseoil-water flow model that utilizes 10 equations to solve for: oil andwater holdups, oil, water, and mixture superficial velocities, mixturedensity and dynamic viscosity, a slip velocity between oil and waterphases, a Reynolds number and Fanning friction factor, and a pressureloss that occurs over a metering length of a flow conduit.
 6. The systemof claim 1, wherein the flow analysis model includes a two-phase gas-oilflow model that utilizes 10 equations to solve for: oil and gas holdups,superficial velocities, a mixture superficial velocity, density andviscosity, a gas-liquid slip velocity, a Reynolds number and a Fanningfriction factor.
 7. The system of claim 1, wherein the flow analysismodel includes a two-phase water-gas flow model that utilizes 10equations to solve for: water and gas holdups, water and gas superficialvelocities, a mixture superficial velocity, density and dynamicviscosity, a gas-liquid slip velocity, a Reynolds number and a Fanningfriction factor.
 8. The system of claim 1, wherein the flow analysismodel includes a three-phase flow model that utilizes 13 equations tosolve for: holdups for oil, water and gas phases, superficial velocitiesof oil, water and gas, a mixture superficial velocity, a mixturedensity, a viscosity, a Reynolds number, a friction factor, a water-oilslip velocity and a gas-liquid slip velocity.
 9. The system of claim 1,wherein said wellbore is used to produce formation fluids or to injectfluids into a formation.
 10. The system of claim 1, wherein saidcomputation system calibrates one or more of said wellbore fluidproperties using data obtained from a retrievable production loggingdevice.
 11. A method for analyzing multiphase flow in a wellbore,comprising: obtaining pressure and temperature readings from a pair ofsensors located in the wellbore; utilizing a flow analysis model togenerate a set of wellbore fluid properties from the pressure andtemperature readings, wherein the set of wellbore fluid propertiesincludes at least one of: a fluid mixture value, a phase velocity value,a flow rate, a mixture density, a mixture viscosity, a fluid holdup, anda slip velocity; and outputting the wellbore fluid properties.
 12. Themethod of claim 11, wherein the pair of sensors are permanently mountedin the wellbore.
 13. The method of claim 11, further comprisinganalyzing pressure and temperature from a plurality of sensor pairs toprovide wellbore fluid properties for different inflow zones and/orbranches of a multiple production zone well.
 14. The method of claim 11,wherein the flow analysis model includes a single-phase flow model thatutilizes three equations to solve for a fluid superficial velocity, aReynolds number, and a Fanning friction factor.
 15. The method of claim11, wherein the flow analysis model includes a two-phase oil-water flowmodel that utilizes 10 equations to solve for: oil and water holdups,oil, water, and mixture superficial velocities, mixture density anddynamic viscosity, a slip velocity between oil and water phases, aReynolds number and Fanning friction factor, and a pressure loss thatoccurs over a metering length of a flow conduit.
 16. The method of claim11, wherein the flow analysis model includes a two-phase gas-oil flowmodel that utilizes 10 equations to solve for: oil and gas holdups,superficial velocities, a mixture superficial velocity, density andviscosity, a gas-liquid slip velocity, a Reynolds number and a Fanningfriction factor.
 17. The method of claim 11, wherein the flow analysismodel includes a two-phase water-gas flow model that utilizes 10equations to solve for: water and gas holdups, water and gas superficialvelocities, a mixture superficial velocity, density and dynamicviscosity, a gas-liquid slip velocity, a Reynolds number and a Fanningfriction factor.
 18. The method of claim 11, wherein the flow analysismodel includes a three-phase flow model that utilizes 13 equations tosolve for: holdups for oil, water and gas phases, superficial velocitiesof oil, water and gas, a mixture superficial velocity, a mixturedensity, a viscosity, a Reynolds number, a friction factor, a water-oilslip velocity and a gas-liquid slip velocity.
 19. The method of claim11, wherein said wellbore is used to produce formation fluids or toinject fluids into a formation.
 20. The system of claim 11, wherein dataobtained from a retrievable production logging device is used tocalibrate one or more of said wellbore fluid properties.
 21. A computerreadable medium for storing a computer program product, which whenexecuted by a computer system analyzes multiphase flow in a wellbore,comprising: program code for inputting pressure and temperature readingsfrom a pair of sensors located in the wellbore; program code forimplementing a flow analysis model to generate a set of wellbore fluidproperties from the pressure and temperature readings, wherein the setof wellbore fluid properties includes at least one of: a fluid mixturevalue, a phase velocity value, a flow rate, a mixture density, a mixtureviscosity, a fluid holdup, and a slip velocity; and program code foroutputting the wellbore fluid properties.
 22. The computer readablemedium of claim 21, wherein the wellbore fluid properties are outputtedin an on-demand manner.
 23. The computer readable medium of claim 21,wherein the pair of sensors are permanently mounted in the wellbore. 24.The computer readable medium of claim 21, further comprising programcode for analyzing pressure and temperature from a plurality of sensorpairs to provide wellbore fluid properties for different branches of amultiple production zone well.
 25. The computer readable medium of claim21, wherein the flow analysis model includes a single-phase flow modelthat utilizes three equations to solve for a fluid superficial velocity,a Reynolds number, and a Fanning friction factor.
 26. The computerreadable medium of claim 21, wherein the flow analysis model includes atwo-phase oil-water flow model that utilizes 10 equations to solve for:oil and water holdups, oil, water, and mixture superficial velocities,mixture density and dynamic viscosity, a slip velocity between oil andwater phases, a Reynolds number and Fanning friction factor, and apressure loss that occurs over a metering length of a flow conduit. 27.The computer readable medium of claim 21, wherein the flow analysismodel includes a two-phase gas-oil flow model that utilizes 10 equationsto solve for: oil and gas holdups, superficial velocities, a mixturesuperficial velocity, density and viscosity, a gas-liquid slip velocity,a Reynolds number and a Fanning friction factor.
 28. The computerreadable medium of claim 21, wherein the flow analysis model includes atwo-phase water-gas flow model that utilizes 10 equations to solve for:water and gas holdups, water and gas superficial velocities, a mixturesuperficial velocity, density and dynamic viscosity, a gas-liquid slipvelocity, a Reynolds number and a Fanning friction factor.
 29. Thecomputer readable medium of claim 21, wherein the flow analysis modelincludes a three-phase flow model that utilizes 13 equations to solvefor: holdups for oil, water and gas phases, superficial velocities ofoil, water and gas, a mixture superficial velocity, a mixture density, aviscosity, a Reynolds number, a friction factor, a water-oil slipvelocity and a gas-liquid slip velocity.
 30. The computer readablemedium of claim 21, further including program code for calibrating oneor more of said wellbore fluid properties using data obtained from aretrievable production logging device.